QR Decomposition
Factor A = QR with orthogonal Q and triangular R.
Key idea
Factor A = QR with orthogonal Q and triangular R.
In depth
Any matrix with independent columns factors as A = QR, with Q orthonormal columns (from Gram–Schmidt) and R upper triangular. It is numerically stable for least squares and eigenvalue algorithms.
Worked example
Solving Ax = b becomes Rx = Qᵀb, solved by back-substitution.
Formula
Real-world uses
- Solving least squares problems stably in statistics software and GPS position estimation
Common mistake
Using classical Gram–Schmidt on ill-conditioned matrices; it loses accuracy.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is QR Decomposition?
Factor A = QR with orthogonal Q and triangular R. It is a topic at the expert level in Linear Algebra.
What is the formula for QR Decomposition?
The key formula is A = QR.
Where is QR Decomposition used in real life?
Real-world examples: Solving least squares problems stably in statistics software and GPS position estimation.
What should I learn before QR Decomposition?
Make sure you are comfortable with Gram–Schmidt Process, the previous topic in Linear Algebra.
What is a common mistake with QR Decomposition?
Using classical Gram–Schmidt on ill-conditioned matrices; it loses accuracy.
How can I practise QR Decomposition?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after QR Decomposition?
Move on to Least Squares next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.